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Properties

Label 2-15-15.2-c11-0-5
Degree $2$
Conductor $15$
Sign $0.936 - 0.351i$
Analytic cond. $11.5251$
Root an. cond. $3.39487$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−48.4 + 48.4i)2-s + (−342. + 244. i)3-s − 2.64e3i·4-s + (−5.08e3 + 4.79e3i)5-s + (4.74e3 − 2.84e4i)6-s + (−4.78e4 − 4.78e4i)7-s + (2.91e4 + 2.91e4i)8-s + (5.75e4 − 1.67e5i)9-s + (1.41e4 − 4.78e5i)10-s + 8.40e5i·11-s + (6.47e5 + 9.07e5i)12-s + (−1.15e6 + 1.15e6i)13-s + 4.64e6·14-s + (5.69e5 − 2.88e6i)15-s + 2.60e6·16-s + (−1.93e6 + 1.93e6i)17-s + ⋯
L(s)  = 1  + (−1.07 + 1.07i)2-s + (−0.813 + 0.581i)3-s − 1.29i·4-s + (−0.727 + 0.685i)5-s + (0.249 − 1.49i)6-s + (−1.07 − 1.07i)7-s + (0.314 + 0.314i)8-s + (0.324 − 0.945i)9-s + (0.0446 − 1.51i)10-s + 1.57i·11-s + (0.751 + 1.05i)12-s + (−0.862 + 0.862i)13-s + 2.30·14-s + (0.193 − 0.981i)15-s + 0.620·16-s + (−0.330 + 0.330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.936 - 0.351i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 0.936 - 0.351i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.144667 + 0.0262334i\)
\(L(\frac12)\) \(\approx\) \(0.144667 + 0.0262334i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (342. - 244. i)T \)
5 \( 1 + (5.08e3 - 4.79e3i)T \)
good2 \( 1 + (48.4 - 48.4i)T - 2.04e3iT^{2} \)
7 \( 1 + (4.78e4 + 4.78e4i)T + 1.97e9iT^{2} \)
11 \( 1 - 8.40e5iT - 2.85e11T^{2} \)
13 \( 1 + (1.15e6 - 1.15e6i)T - 1.79e12iT^{2} \)
17 \( 1 + (1.93e6 - 1.93e6i)T - 3.42e13iT^{2} \)
19 \( 1 - 1.46e6iT - 1.16e14T^{2} \)
23 \( 1 + (8.35e6 + 8.35e6i)T + 9.52e14iT^{2} \)
29 \( 1 + 2.19e7T + 1.22e16T^{2} \)
31 \( 1 - 3.79e7T + 2.54e16T^{2} \)
37 \( 1 + (-2.67e8 - 2.67e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 4.76e8iT - 5.50e17T^{2} \)
43 \( 1 + (-6.84e8 + 6.84e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (-7.25e8 + 7.25e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (1.12e9 + 1.12e9i)T + 9.26e18iT^{2} \)
59 \( 1 - 1.04e9T + 3.01e19T^{2} \)
61 \( 1 + 3.61e9T + 4.35e19T^{2} \)
67 \( 1 + (-1.85e9 - 1.85e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.66e10iT - 2.31e20T^{2} \)
73 \( 1 + (-3.27e9 + 3.27e9i)T - 3.13e20iT^{2} \)
79 \( 1 + 4.46e10iT - 7.47e20T^{2} \)
83 \( 1 + (-1.43e10 - 1.43e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 7.02e10T + 2.77e21T^{2} \)
97 \( 1 + (-4.68e10 - 4.68e10i)T + 7.15e21iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.74724467471424348315624081217, −15.75348068055017041817976815993, −14.76618958829096569748148964998, −12.27392451298650619115241104903, −10.39968755275294258599921515452, −9.600524374483527431691455514749, −7.27976734346179618703002196853, −6.63243506134409168478871437662, −4.15565995050591490810498449561, −0.17175064353450946876470972228, 0.71811170177800412542502863356, 2.82744587702577379929978981937, 5.73250950137656020528575728996, 8.029355920235290587681197561254, 9.352587468751329491301161567218, 11.04490315554231774073923251838, 12.07307868093002661941014524180, 12.92287578552938187988004072924, 15.82420984173885126864970611536, 16.85833682042440627369464723812

Graph of the $Z$-function along the critical line